Embedded newform 224.3.w.a.43.8

• Label: 224.3.w.a.43.8
• Level: $224$
• Weight: $3$
• Character: 224.43
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.w (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			43.8
Character	\(\chi\)	\(=\)	224.43
Dual form			224.3.w.a.99.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.75066 - 0.967050i) q^{2} +(-1.86844 - 4.51082i) q^{3} +(2.12963 + 3.38595i) q^{4} +(0.599529 - 1.44739i) q^{5} +(-1.09118 + 9.70379i) q^{6} +(-1.87083 + 1.87083i) q^{7} +(-0.453865 - 7.98712i) q^{8} +(-10.4924 + 10.4924i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.75066	−	0.967050i	−0.875330	−	0.483525i
							\(3\)	−1.86844	−	4.51082i	−0.622814	−	1.50361i	−0.848385	−	0.529380i	\(-0.822425\pi\)
0.225571	−	0.974227i	\(-0.427575\pi\)
\(5\)	0.599529	−	1.44739i	0.119906	−	0.289478i	−0.852518	−	0.522697i	\(-0.824926\pi\)
							0.972424	+	0.233219i	\(0.0749259\pi\)
\(7\)	−1.87083	+	1.87083i	−0.267261	+	0.267261i
							\(11\)	−8.17903	+	19.7459i	−0.743548	+	1.79508i	−0.152748	+	0.988265i	\(0.548812\pi\)
−0.590800	+	0.806818i	\(0.701188\pi\)
\(13\)	1.00363	+	2.42299i	0.0772027	+	0.186384i	0.957769	−	0.287540i	\(-0.0928373\pi\)
							−0.880566	+	0.473923i	\(0.842837\pi\)
\(17\)		−	22.9978i		−	1.35281i	−0.736529	−	0.676406i	\(-0.763537\pi\)
							0.736529	−	0.676406i	\(-0.236463\pi\)
\(19\)	−28.2373	+	11.6963i	−1.48617	+	0.615593i	−0.970480	−	0.241180i	\(-0.922465\pi\)
							−0.515693	+	0.856774i	\(0.672465\pi\)
\(23\)	19.0036	+	19.0036i	0.826244	+	0.826244i	0.986995	−	0.160751i	\(-0.0513916\pi\)
							−0.160751	+	0.986995i	\(0.551392\pi\)
\(29\)	23.1121	−	9.57336i	0.796971	−	0.330116i	0.0532282	−	0.998582i	\(-0.483049\pi\)
							0.743742	+	0.668466i	\(0.233049\pi\)
\(31\)		−	12.8494i		−	0.414497i	−0.978288	−	0.207249i	\(-0.933549\pi\)
							0.978288	−	0.207249i	\(-0.0664509\pi\)
\(37\)	−17.0724	+	41.2165i	−0.461417	+	1.11396i	0.506398	+	0.862300i	\(0.330977\pi\)
							−0.967816	+	0.251660i	\(0.919023\pi\)
\(41\)	16.4556	−	16.4556i	0.401357	−	0.401357i	−0.477354	−	0.878711i	\(-0.658404\pi\)
							0.878711	+	0.477354i	\(0.158404\pi\)
\(43\)	−9.75224	+	23.5440i	−0.226796	+	0.547535i	−0.995784	−	0.0917288i	\(-0.970761\pi\)
							0.768988	+	0.639264i	\(0.220761\pi\)
\(47\)	−40.5701			−0.863194			−0.431597	−	0.902066i	\(-0.642050\pi\)
							−0.431597	+	0.902066i	\(0.642050\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.43.8	✓	192
32.3	odd	8	inner	224.3.w.a.99.8	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.8	✓	192	1.1	even	1	trivial
224.3.w.a.99.8	yes	192	32.3	odd	8	inner